Rational and Irrational Numbers

Warm-up

Draw a unit square and its diagonal: "The sides are 1. How long is the diagonal — exactly?" Pythagoras gives $\sqrt{2}$. Now the provocation: "Fine — write $\sqrt{2}$ as a fraction." Let them try 1.4 ($\frac{7}{5}$: squares to 1.96), 1.41, $\frac{99}{70}$ (squares to 2.0002…) — every candidate misses.

The reveal: no fraction EVER hits it. Numbers exist that are not ratios of integers — irrationals — and one is hiding in every square's diagonal.

Explore

Number-system census: sort a zoo of numbers ($\frac{3}{4}$, $0.\overline{7}$, $\sqrt{25}$, $\sqrt{10}$, $\pi$, $-6$, $0.121121112\ldots$) into rational vs irrational, with the operational test: rational = expressible as $\frac{a}{b}$ = decimal that terminates OR repeats. The engineered decimal $0.121121112\ldots$ (pattern grows, never repeats) earns its irrational badge by structure.

Then locate irrationals physically: construct $\sqrt{2}$ on a number line with compass-and-diagonal (swing the unit square's diagonal down to the axis) — an irrational number with an exact ADDRESS, pinned by geometry that decimals can only approximate.

Formalize

Formalize the hierarchy — naturals inside integers inside rationals, with irrationals as the rest of the real line:

$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \qquad \mathbb{Q} \cup \text{irrationals} = \mathbb{R} \qquad \sqrt{2} \notin \mathbb{Q}$

The decimal trichotomy that decides membership: terminating (rational), repeating (rational), neither (irrational). Roots sort quickly: $\sqrt{n}$ is rational exactly when $n$ is a perfect square — everything between squares is irrational, so irrationals are EVERYWHERE (between any two squares, infinitely many).

Practice

Practice: classify twelve numbers with reasons; convert $0.\overline{45}$ to a fraction (the ×100-and-subtract trick: $x = 0.\overline{45}$, $100x = 45.\overline{45}$, $99x = 45$, $x = \frac{5}{11}$); order a mixed rational/irrational set using root-trapping; one construction of $\sqrt{5}$ on the line (legs 1 and 2).

Exit ticket: is $\frac{\sqrt{50}}{\sqrt{2}}$ rational or irrational? (Simplify first: $\sqrt{25} = 5$ — rational! Irrational ingredients can cook rational meals.)

Exit ticket

Practice: classify twelve numbers with reasons; convert $0.\overline{45}$ to a fraction (the ×100-and-subtract trick: $x = 0.\overline{45}$, $100x = 45.\overline{45}$, $99x = 45$, $x = \frac{5}{11}$); order a mixed rational/irrational set using root-trapping; one construction of $\sqrt{5}$ on the line (legs 1 and 2).

Exit ticket: is $\frac{\sqrt{50}}{\sqrt{2}}$ rational or irrational? (Simplify first: $\sqrt{25} = 5$ — rational! Irrational ingredients can cook rational meals.)